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Graphs & Linear Equations
Y
X
Example of a Linear Function
A Dog’sHuman’sEquivalentAge
A Dog’s Actual Age
Y
X
(3,21)
(5,35)
(11,77)•
•
•
3 5 11
77
35
21
(0,0)
y=f(x)=7x
Major Elements of Graphing Lines
• Graphing Ordered Pairs
• Graphing Equations
• Linear Equations
• Slope & Equations
• Finding Equations of Lines
• Fitting Equations to Lines
• Parallel & Perpendicular Lines
A PointY
X
(X,Y) is called an Ordered Pair
The X value or X Coordinate is the location of a point in the X direction
The Y value or Y Coordinate is the location of a point in the Y direction
(X,Y) (4,3) •
How to Graph a PointY
X
(X,Y) (4,2) •
X is the distance along the x=axis
Y is the distance along the y=axis
1 2 3 4
321
HINT: Think of the x-axis as the Number Line
-4-3-2-1 0
X1 2 3 4-4-3-2-1 0
HINT: Think of the y-axis a vertical Number Line
Y
3210
-1-2
Important Vocabulary for Graphs
Y
X
The Graph itself is called the x-y plane (ie. Plane surface) or The Coordinate Plane or Cartesian Coordinate Plane after Renee Descartes
(+X,+Y)
(+X,-Y)(-X,-Y)
(-X,+Y)
Graphing Linear Equations(Find 3 Domain & Range Points)
First Degree Equations are Lines (y=mx+b) and you calculate 3 (X,Y) values
Make sure the points line up on a x-y graph and connect the dots.
RECALL X-Domain & Y-Range
y = 2x-7 when the Domain is {-2, 0, 2}
f(-2) = 2•(-2) -7 = -4 -7 = -11 (-2,-11)f(0) = 2•(0) -7 = 0 -7 = -7 (0,-7)f(2) = 2•(2) -7 = 4 -7 = -3 (2,-3)
Answer: RANGE: {-11, -7, -3}
Graphing Lines is just like finding the Range of 3 Domain Points:(Substitute each Domain value into the equation)
Practice Finding 3 PointsGiven a Linear Equation
Find any 3 (X,Y) points for the following equations:
y=5x
y=4x-5
y=3x+1
(Hint: Try x=0)
Sample Solutions
x y = 5x
0 0
1 5
2 10
x y = 4x-5 x y = 3x+1
Now Graph the 3 Pointsx y = 5x
0 0
1 5
2 10
•
•
•(0,0)
(1,5)
(2,10)
What is Intercept in Math?Y
X
An Intercept is the coordinate where a line crosses the x or y axis
Using X&Y Intercepts to Graph a Line
The X intercept is the x coordinate (where a line crosses the x axis).
Y
X
(0,2)
(3,0)
••
The Y intercept is the y coordinate (where a line crosses the y axis).
Name the X&Y Intercepts
Y
X
(0,2)
(3,0)
••
Name the X&Y Intercepts
Y
X
(0,2)
(-2,0)
••
Name the X&Y Intercepts
Y
X
Name the X&Y Intercepts
Y
X
What is the value of x at the y intercept? What is the value of Y at the x-intercept?
Y
X
Graph y = 2x - 6 using x&y intercepts
1st Make x-y table
X Y = 2x - 6
Y
X
Graph Linear Eq.
Graph y = 2x - 6 using x&y intercepts
1st Make x-y table
2nd Set x = 0 and solve for y
X Y = 2x - 6
0 -6
Y
X
• (0,-6)
Graph Linear Eq.
Graph y = 2x - 6 using x&y intercepts
1st Make x-y table
2nd Set x = 0 and solve for y
3rd Set y = 0 and solve for x
X Y = 2x - 6
0 -6
3 0Y
X
• (0,-6)
• (3,0)
Graph Linear Eq.
Graph y = 2x - 6 using x&y intercepts
1st Make x-y table
2nd Set x = 0 and solve for y
3rd Set y = 0 and solve for x
4th Plot these 2 points and draw line
X Y = 2x - 6
0 -6
3 0Y
X
• (0,-6)
• (3,0)
Graph Linear Eq.
Graph y = 2x - 6 using x&y intercepts
1st Make x-y table
2nd Set x = 0 and solve for y
3rd Set y = 0 and solve for x
4th Plot these 2 points and draw line
5th Use 3rd point to check
X Y = 2x - 6
0 -6
3 0
4 2Y
X
• (0,-6)
• (3,0)• (4,2)
Graph Linear Eq.
Graphing Horizontal & Vertical Lines
Y
X
This line has a y value of 4 for any x-value. It’s equation is
y = 4 (meaning y always equals 4)
Graphing Horizontal & Vertical Lines
Y
X
This line has a x value of 1 for any y-value. It’s equation is
x = 1 (meaning x always equals 1)
The Equation of a Vertical Line is X=Constant
Y
X
x = 1
The Equation of a Horizontal Line is Y=Constant
Y
X
y = 3
Graph the following lines
Y = -4
Y = 2
X = 5
X = -5
X = 0
Y = 0
Answers
Y
X
x = 5x = -5
Answers
Y
X
y = -4
y = 2
Answers
Y
X
x = 0
y = 0
SLOPE =
Slope is a measure of STEEPNESS
RISERUN
The Symbol forSLOPE = m
Think of m for Mountain
SLOPE =
RISERUN
RUN
RISE•
•
How much does this line rise?
How much does it run?
(3,2)
(6,4)
(0,0) 1 2 3
12
4
3
5 6
4
SLOPE =
RISERUN
•
•
How much does this line rise?
How much does it run?
(3,2)
(6,4)
(0,0) 1 2 3
12
4
3
5 6
4
RUN 3
RISE 2
m=SLOPE =
•
•
(3,2)
(6,4)
(0,0) 1 2 3
12
4
3
5 6
4
RUN 3
RISE 2
Slopemy2 y1
x2 x1
4 26 3
23
RISERUN
y2 y1
x2 x1
x1y1
x2y2
•
•
(x2,y2)(6’4)
(x1,y1)(3,2)
Switch points and calculate slopeMake (3,2) (x2,y2) & (6,4) (x1,y1)
•
•
(x1,y1)(6,4)
(x2,y2)(3,2)
Recalculation with points switched
Slopemy2 y1
x2 x1
4 62 5
2 3
23
•
•
(x1,y1)(6,4)
(x2,y2)(3,2)
Same slope as before
It doesn’t matter what 2 points you choose on a line
the slope must come out the same
Keeping Track of Signs When Finding The Slope Between 2 Points
• Be Neat & Careful
• Use (PARENTHASES)
• Double Check Your Work as you Go
• Follow 3 Steps
3 Steps for finding the Slope of a line between 2 Points
(3,4)&(-2,6)
Slopey2 y1
x2 x1
6 4 2 3
1st Step: Write x1,y1,x2,y2 over numbers
2nd Step: Write Formula and Substitute x1,x2,y1,y2 values.
3rd Step: Calculate & Simplify
(3,4) & (-2,6)x1 y1 x2 y2
6 4 2 3
2 5
25
Find the Slopes of Lines containing these 2 Points
1. (1,7) & (5,2)
5. (3,6) & (5,-5)
3. (-3,-1) & (-5,-9)
6. (1,-4) & (5,9)
4. (4,-2) & (-5,4)
2. (3,5) & (-2,-8)
1. (1,7) & (5,2)
5. (3,6) & (5,-5)
3. (-3,-1) & (-5,-9)
6. (1,-4) & (5,9)
4. (4,-2) & (-5,4)
2. (3,5) & (-2,-8)
Slopey2 y1
x2 x1
2 75 1
54
Slopey2 y1
x2 x1
8 5 2 3
13 5
135
Slopey2 y1
x2 x1
9 ( 1) 5 ( 3)
8 2
41
Slopey2 y1
x2 x1
5 65 3
112
Slopey2 y1
x2 x1
4 ( 2) 5 4
6 9
23
Slopey2 y1
x2 x1
9 ( 4)
5 1
134
ANSWERS
Solve for y if (9,y) & (-6,3) & m=2/3
Slopey2 y1
x2 x1
23
3 y1
6 9
3 y 15
( 15)23
3 y1
6 9
3 y 15
( 15)
( 5)2 3 y
10 3 y
13 y
13 y
Review Finding the Slopes of Lines Given 2 Points
1st Step: Write x1,x2,y1,y2 over numbers
2nd Step: Write Formula and Substitute x1,x2,y1,y2 values.
3rd Step: Calculate & Simplify
NOTE:
Be Neat, Careful, and Precise and Check your work as you go..
mSlopey2 y1
x2 x1
SLOPE mRISERUN
ZERO Slope Horizontal
Positive SlopeIs Up the Hill
Negative SlopeIs Down the Hill
NO SlopeVertical Drop
SLOPE mRISERUN
ZERO Slope Horizontal
NO SlopeVertical Drop
RISERUN
0
any _ number0
RISERUN
any _ number
0Undefined(NO_ Slope)
Equations of a LineThere are 3 Forms of Line Equations
• Standard Form: ax+by=c
• Slope Intercept Form: y=mx+b
• Point-Slope Form y-y1=m(x-x1)
All 3 describe the line completely but are used for different purposes. You can convert from one form to another.
Converting fromStandard Form: ax+by=c
to Slope Intercept Form
3x 6y 12
6y 3x 12
66y
36x
126
y 12x 2 Slope Intercept Form:
y=mx+b
JUST SOLVE FOR Y
Slope Intercept Form: y=mx+b
The great thing about this form is b is the y-intercept.
This makes graphing a line incredibly easy. Check it out. If
• (0,1)
y 23 x 1
The y intercept is +1Almost a free point on graph
Slope Intercept Form: y=mx+b
All you have to do now is use the slope to rise and run
from the intercept & connect the points.
•(0,1)
y 23 x 1
m riserun
23
•
Rise 2 and Run 3 from the y-intercept & connect points.
y=mx+b when m is negative
All you have to do now is use the slope to rise and run
from the intercept & connect the points.
•(0,1)
y 23 x 1
mriserun
23
•
Rise -2 and Run 3 from the y-intercept & connect points.
Slope Intercept Form: y=mx+bGRAPH THESE LINEAR EQUATIONS
y 12x 1
y 25x 3
y 12x 1
y 32x 1
Label y-intercept & Use one big graph
If linear equation is not in y=mx+b form solve for y
22y
52x
42
y 52x 2
2y 5x 4
Now it isThis line has an y intercept of -2 and rises 5 and runs 2.
Solution Steps to Solve for y:
Divide by 2
Graphing a line with
slope intercept equation
22y
52x
42
y 52x 2
2y 5x 41. Solve for y:
2. Y-Intercept is 1st Point.
3. From the y-intercept
Rise 5 and run 2 for
Second Point.
4. Connect Points with line.
2y 5x 4Graph
Y
X
•(0,-2)
5{2•
y 52x 2 Now it is easy to graph
• (0,-2)
•
Put into slope-intercept form and graph
3y 9x 3
4y 8x 4
y 5 6x
2y 4 6x 2
Review Steps of Graphing from the Slope Intercept Equation
1. Make sure equation is in y=mx+b form
2. Plot b(y-intercept) on graph (0,b)
3. From b, Rise and Run according to the slope to plot 2nd point.
4. Check sign of slope visually
Find the Equation of a Line(Given Pt. & Slope)
Given a point (2,5) & m=5 Write the Equation
y mx b5 5(2) b5 10 b 5 b
y 5x 5
1. Write Slope-Intercept Equation
2. 2. Plug-in (x,y) & m values
3. Solve for b
4. Plug m & b into Slope-Int. Eq.
Find the Equation of a Line(Given Pt. & Slope) Method 2
Using the Pt.-Slope Eq. Given a point (2,5) & m=5 Write the Equation
y y1 m(x x1)y 5 5(x 2)y 5 5x 10y 5x 5
y 5x 5
1. Write Pt.-Slope Equation
2. 2. Plug-in (x,y) & m values
3. Solve for y
Find the Equation of a Line(Given 2 Points)
Given a point (x1,y1) & (x2,y2)
(2,5) & (3,10)
y mx b5 5(2) b5 10 b 5 b
y 5x 5
1. Find Slope using
2. Write Slope-Intercept Equation
3. Plug-in (x,y) & m values
4. Solve for b
5. Plug m & b into Slope-Int. Eq.
mSlopey2 y1
x2 x1
Parallel LinesHave the Same Slope
RUN 3
RISE 2•
•
(0,0) 1 2 3
12
4
3
5 6
45
RUN 3
RISE 2
Perpendicular LinesHave Neg. Reciprocal Slopes
(0,0) 1 2 3
1
2
4
3
5 6
m1 23
m2 32
m1 m2 23
32
1
Systems of Equations
Given 2 linear equations
The single point where they intersect is a solution to either equation
It is also the solution to both equations or what we call the solution to the SYSTEM OF EQUATIONS
•
(0,0) 1 2 3
1
2
4
3
5 6
y x 1m2 1
y x 3m1 1
-1
• Solution(0,-1)
(0,3)
(2,1)
(0,0) 1 2 3
1
2
4
3
5 6
y x 1m2 1
y x 3m1 1
-1
• Solution(0,-1)
(0,3)
(2,1)
Systems of Equations
The Solution is where the two lines meet (or intersect)
(0,0) 1 2 3
1
2
4
3
5 6
y x 1m2 1
y x 3m1 1
-1
• Solution(0,-1)
(0,3)
(2,1)